Homography Matrix Calculation Using Triangle Area Ratios
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Solution Overview
Problem
Existing methods for calculating homography and projective transformation matrices are computationally intensive and prone to numerical errors, especially when dealing with large numbers of feature points in applications like vibration-proofing video cameras, where high-speed matrix deduction is necessary.
Innovation Solution
An information processing apparatus that calculates transformation matrices by determining area or volume ratios of triangles or tetrahedrons formed by feature points, simplifying the arithmetic operations and reducing the computational load, and includes mechanisms to quickly determine linearity or planarity to abort unnecessary matrix deduction processes.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If Gauss iteration method or Gauss-Jordan iteration method is used to solve simultaneous equations for homography matrix, then the homography matrix can be obtained, but the operation amount becomes large and numerical calculation errors increase
Solution Approach 1:
The patent divides the homography matrix calculation into two independent stages: first calculating transformation matrices from reference coordinates to feature points in each image separately, then combining them. This segmentation reduces the complexity of solving large simultaneous equations directly, thereby reducing operation amount while maintaining accuracy.
Solution Approach 2:
The patent introduces a unit square as an intermediary reference frame. Instead of directly solving for the homography matrix between two images, the calculation is mediated through the unit square, which simplifies the arithmetic operations and reduces computational complexity while maintaining the transformation accuracy.
2Measurement precision
If pivot method is used to eliminate numerical errors in simultaneous equations, then calculation accuracy improves, but the processing amount further increases
Solution Approach 1:
The patent segments the calculation process into separate transformation matrix computations from reference coordinates to feature points in each image, followed by matrix multiplication. This avoids the need for pivot operations in solving large simultaneous equations, reducing processing amount while maintaining accuracy.
Solution Approach 2:
The patent replaces the mechanical iterative solving process (Gauss iteration, Gauss-Jordan elimination, pivot methods) with a direct matrix-based approach using area ratios and matrix multiplication, which is computationally more efficient and avoids numerical instability issues.
3Measurement precision
If homography matrices are calculated for as many feature point combinations as possible to obtain highly evaluated matrix, then the evaluation quality improves, but the calculation time increases significantly
Solution Approach 1:
The patent performs preliminary calculation of transformation matrices from reference coordinates to feature points in each image separately. This preliminary action can be reused for multiple feature point combinations, significantly reducing the time required to calculate homography matrices for numerous combinations without sacrificing evaluation quality.
Solution Approach 2:
The patent uses the unit square as a standardized reference template. By calculating transformation matrices from this standard reference to feature points in each image, the same reference framework can be reused across different feature point combinations, reducing redundant calculations and saving time.
4Productivity
If linearity determination process is added to abort matrix deduction when four feature points have linearity, then unnecessary calculations are avoided, but the processing complexity increases
Solution Approach 1:
The patent calculates the area of triangles formed by feature points as a preliminary check before proceeding with matrix deduction. This preliminary action quickly identifies linear configurations (where area is zero or negligible) and allows early termination of unnecessary calculations, improving efficiency without significant complexity increase.
Solution Approach 2:
The patent replaces complex linearity determination algorithms with a simple geometric area calculation approach. By computing the area of triangles formed by feature points, the system can quickly and efficiently determine if points are collinear, avoiding the need for more complex linear independence tests while maintaining accuracy.
Data Source
AI summary
When deducing a homography matrix representing a geometrical relationship between two images, an image processing apparatus obtains a transformation matrix for four feature points in each of the two images based on reference coordinates. More specifically, the apparatus obtains the area of a triangle formed by three out of the four feature points, and calculates projective parameters of transformation matrices from the area ratios. The apparatus calculates the homography matrix between the images by the product of one of the two obtained matrices and an inverse matrix of the other.